Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13669 |
| automorphism | Aut(G) = D4×#12067 (order 2048) · Inn(G) = #2184 · Out(G) = C2×C8 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2184 |
| commutator, Frattini, cosocle | C2×C16 | C23 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #504 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
144 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,16)(4,21)(5,14)(6,30)(7,12)(8,15)(9,17)(10,18)(11,25)(13,23)(19,50)(20,43)(22,45)(24,33)(26,52)(27,53)(28,38)(29,46)(31,39)(32,44)(34,55)(35,54)(36,49)(37,51)(40,57)(41,58)(42,59)(47,48)(56,64)(60,62)(61,63);(1,3)(2,9)(4,22)(5,18)(6,31)(7,28)(8,20)(10,33)(11,39)(12,34)(13,45)(14,47)(15,36)(16,21)(17,25)(19,23)(24,54)(26,60)(27,56)(29,51)(30,35)(32,61)(37,41)(38,52)(40,50)(42,63)(43,53)(44,62)(46,64)(48,58)(49,59)(55,57);(1,3)(2,10)(4,17)(5,28)(6,19)(7,18)(8,20)(9,33)(11,35)(12,43)(13,37)(14,16)(15,38)(21,47)(22,25)(23,31)(24,55)(26,48)(27,49)(29,61)(30,39)(32,51)(34,53)(36,52)(40,64)(41,45)(42,62)(44,63)(46,50)(54,57)(56,59)(58,60)Polycyclic presentation:
2^8|d^2=fg,e^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fg,g^a=gh,d^b=df,f^b=fg,g^b=gh,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13669, #512848, #512849, … |
| maximal normal subgroup of | C2×#13669, #512848, #512849, … |
| maximal quotient of | #151176, #151178, #151288, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6657);